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Edited by: Sergey M. Korogod, National Academy of Sciences of Ukraine, Ukraine

Reviewed by: Benjamin Torben-Nielsen, Okinawa Institute of Science and Technology, Japan; Malte J. Rasch, Beijing Normal University, China

*Correspondence: Matthew F. Singh, The Program in Neurosciences, 660 S. Euclid Ave., St. Louis, MO 63110, USA

David H. Zald, Department of Psychology, Vanderbilt University, PMB 407817, Nashville, TN 37240-7817, USA

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Relatively recent advances in patch clamp recordings and iontophoresis have enabled unprecedented study of neuronal post-synaptic integration (“dendritic integration”). Findings support a separate layer of integration in the dendritic branches before potentials reach the cell's soma. While integration between branches obeys previous linear assumptions, proximal inputs within a branch produce threshold nonlinearity, which some authors have likened to the sigmoid function. Here we show the implausibility of a sigmoidal relation and present a more realistic transfer function in both an elegant artificial form and a biophysically derived form that further considers input locations along the dendritic arbor. As the distance between input locations determines their ability to produce nonlinear interactions, models incorporating dendritic topology are essential to understanding the computational power afforded by these early stages of integration. We use the biophysical transfer function to emulate empirical data using biophysical parameters and describe the conditions under which the artificial and biophysically derived forms are equivalent.

Over the past decade, increasing evidence indicates that dendritic architecture plays an active role in shaping somatic responses to synaptic input. Particularly in pyramidal neurons (e.g., Schiller et al., ^{+}, Ca^{+}, and/or NMDA spikes (Schiller et al.,

In contrast to the oft described “sigmoid,” Poirazi et al. (

We begin by characterizing the dendritic transfer function T_{D}(V):

As the distance between input sites increases,

At close distances,

Three currents must be accounted for: fast ionic currents (I_{fast}), leak current (I_{leak}), and a slow NMDAR-mediated current (I_{NMDA}).

Biophysical models have made use of the fast-slow dynamics of dendritic membrane to neglect relaxation times of fast channels, instead keeping them constant at equilibrium conductance (Genet and Delord, _{D}(x_{1}, x_{2}, x_{3},…)] using the distances between input sites as a proxy for time in determining an expectation for leak and NMDAR-mediated currents. The change in potential (relative to base) is then expressed as a bounded sum of linear inputs and nonlinear NMDAR-mediated currents.

As previously mentioned, the sigmoid function does not converge to the linear summation observed for inter-branch dendritic currents. Instead we make use of a juxtaposition of sigmoid integrals of the total polarization to form a locally linear function _{u}_{l}.

Here α_{L} and α_{U} are the curvature of lower and upper boundaries respectively, while _{L} and _{U} are the lower and upper boundaries with the constant _{L} added to center the function (Figure _{Leak} = 0 for the leak potential. Using the multivariate logistic-sigmoid: σ(X_{D}):R^{n} → R = [1 + exp(−∑{X_{i}})]^{−1} for input vector _{D} we first describe a simple transfer function which, as can be seen in Figure

This naïve form simply takes the boundary of the sum of linear and sigmoidally-nonlinear components from the dendritic input vector _{d} the nonlinear maximum, _{d} the curvature, and _{d} the mid-point (related to threshold) of the nonlinear component (Figure

To approximate peak EPSP amplitude as a function of input only, we make use of hierarchical dendritic time scales with the separation principle. In this approach, systems with slow and fast components are separated into a fast subsystem, in which the slow variables are held constant, and a slow subsystem contained in the fast nullcline. In the current case, the “linear” fast ionic currents stem from channels with substantially shorter opening times than NMDAR's, while the opening of NMDAR's and Mg^{2+} unblocking is many orders quicker than channel closing (Jahr and Stevens, _{i}) as in neurotransmission and brief current pulses. Throughout, vectors are ordered from the least to most distal dendritic segments and all potentials are translated for a resting potential of zero.

Because the fast ionic current is propagated passively, we consider it subject to spatial decay only. Decay is characterized by the functional length constant (λ), which is an empirical parameter derived by fitting attenuation data to a negative exponential of distance. Hence the attenuation from spatial decay, denoted φ(_{j→i}

The functional length constant should not be confused with Rall's (

For simplicity, we divide the NMDAR system into binary open and closed phases and take expectations based on open/close time distributions to transition between phases (Figure

Here _{i} are the time constants of each exponential, θ_{i} the associated amplitudes, and the product _{m}_{ON}

This matrix is symmetric with diagonals corresponding to the constants of temporal decay while nondiagonals correspond to the constants of spatial decay when currents propagate from more distal locations(

As stated previously, the NMDAR system may be separated into slow (closing/current flow) and fast (opening/Mg^{2+} gating) subcomponents. Because the fast subsystem rapidly converges to the steady state, the gate's nullcline is stable while the channel is open with nullcline:

Due to the strong time separation, we follow the tradition of considering the gating function to be instantaneous, hence defined by the nullcline (Jahr and Stevens,

Here _{s}), less this reduction lead to global stability while the channel is open. Global stability would compromise the voltage dependence of spike production as glutamate binding would always result in an NMDA spike. Depending on parametrization, Equation (9) may have up to three equilibria, enabling bistability. Equilibria correspond to solutions of the implicit equation:

In the case of three equilibria, solutions possess locally stable lower (resting potential) and upper (saturation) equilibria. The middle equilibrium in this case is unstable leading to the “all or none” bifurcation in spikes. The single equilibrium case, in contrast, produces global stability, usually near the NMDAR reversal potential. As such, the single equilibrium case is pathological in the absence of other modulating voltage gated cation channels (VGC's) as glutamate binding would almost always produce an NMDA spike. However, in biological conditions, NMDA spikes still approach the NMDAR reversal potential, so the locally stable equilibria are roughly preserved. As long as three equilibria are maintained in the reduction to only leak and NMDAR dynamics, the locally stable equilibria remain accurate. To produce three equilibria, we simply modify the slope of Mg^{2+} blockade to compensate for the nonlinearity lost in removing other VGC's. However, it is important to note that full high-dimensional models include other equilibria due to Na^{+}-spikelet's and, in the apical dendrite, Ca^{2+} spikes (see Antic et al., ^{+} (VGSC's) and Ca^{+} channels (VGCC's) as revealed with application of Na^{+} blocker TTX and Ca^{2+} blocker cadmium (Schiller et al., ^{2+} blockade. In the results section we describe when modification is and is not necessary due to the non-uniform distribution of spike thresholds (Major et al.,

We make further reductions through the bifurcation of solutions. In assessing temporal dynamics after the initial channel opening, we consider the long time course of NMDAR bursts and clustering which give rise to macroscopic currents rather than the brief individual open durations Both decay and spiking occur on far shorter time scales than bursts, so states just prior to closing are almost binary and represent the nonlinear component of peak EPSP. As with the opening of individual channels, the population burst duration is considered multi-exponentially distributed (Gibb and Colquhoun, _{NMDA}):
_{i} the amplitude of the exponential component with slope _{i}. In present form, however, both the local dynamics (Equation 8) and expected NMDA component (Equation 10) lack explicit solutions in terms of ordinary functions. Using the bifurcation, we approximate Equations (8) and (10) by making Mg^{2+} blockade constant, following channel opening. In a fully dynamic regime, this method would not be justified. However, because we are only interested in which equilibria solutions approach, rather than how they get there, this method has fair accuracy, provided the earlier condition that Equation (9) has three solutions. As the slope of Mg^{2+} blockade increases (as was done to ensure bistability), the bifurcation point approaches the midpoint of B(V) (Equation 8). At the same time B(V) approaches a step function. The result is that the Mg^{2+} blockade approaches invariance except for an increasingly small region about _{mid}. Provided a sufficiently small ^{2+} blockade may then be approximated as invariant for initial points at channel opening sufficiently far from _{mid}. Changing the dynamic _{0}

Hence with Mg^{2+} blockade constant while the channels are open Equation (10) becomes linear, so all solutions exponentially approach an equilibrium determined by the Mg^{2+} blockade at channel opening. We stress that this approach is only valid in approximating the path toward an equilibrium for Equation (8), with bistability induced by increasing the Mg^{2+} blockade slope. Because the transfer function only considers peak EPSP, this approach is sufficient for the current purposes but is not a valid approximation for the time course of fully dynamic dendrites. As an exponential, this equation is readily combined with burst length distributions. For a given starting potential, the ending potential with an n-exponential burst length distribution is itself n-exponentially distributed following translation. The expected value used in computing peak EPSP is:

While we only present the case for multi-exponential closing distributions, the expected value is relatively insensitive to the type of distribution chosen as spiking and decay occur on much shorter time scales than the fall of NMDAR-mediated currents. When the NMDAR burst/cluster durations are considered sufficiently long, Equations (11) and (12) simplify to a simple sigmoid as in the artificial transfer function's non-linear component in Equation (2):

The second Equation (13.2) results from substituting the Mg^{2+} blockade Equation (7) and is the same as the equilibria Equation (9) when the Mg^{2+} blockade is assumed invariant between initial depolarization and its limiting equilibrium (spike or rest). As discussed previously, the number of starting points (initial depolarizations) for which this assumption is justified increases with the slope of Equation (7) (inversely proportional to _{s}). Hence, as Mg^{2+} blockade becomes increasingly binary, Equation (12) becomes an increasingly accurate description of NMDAR bistability. When the burst/cluster lengths are further assumed sufficiently long to approach limiting states (spike or rest), Equation (12) reduces to Equations (13). This reduction is greatly desirable as Equations (13) do not require explicit knowledge of burst length distributions.

In generating a (time-independent) transfer function, we sacrifice some information concerning the interaction of nonlinear components (NMDAR currents) in separate dendritic segments. Regardless of the number of incoming spikes, for instance, the induced somatic voltage would not be expected to significantly exceed the NMDAR reversal potential. Due to the continuous distribution of NMDAR's along the path of propagation, surplus depolarization would leak back through NMDAR channels before ever reaching the soma. However, the time independence of a transfer function prohibits fully dynamical propagation. While no individual spike crosses the NMDAR reversal potential using the method described above, summation of multiple spikes may, necessitating a boundary function as in Equation (1) to mimic dendritic saturation. Although the function _{Bio}

Here λ is the functional length constant as in Equation (3), _{NMDA}_{i}

Here

Throughout, parametrizations were generally that of Behabadi and Mel (_{m} = 10 KΩcm^{2}, _{Leak} = −70 mV (translated to 0 mV), _{NMDA} = 0 mV (translated to 70 mV), _{(NMDA)} = 3.9 nS. However, we used the conventional C = 1 μF/cm^{2} as opposed to Behabadi and Mel's unusually large capacitance of twice that much. The mid potential for NMDAR's was _{s} = −23.7 mV (translated to 46.3 mV, Jadi et al., _{s} (2.5 vs. 12.5 mV, Major et al.,

To test the biophysical transfer function, we performed two sets of simulations custom-coded in MatLab2015a (Mathworks Inc., Natick, MA). In the first set of simulations we used a 5-state kinetic model of NMDAR's (Destexhe et al.,

In the second set of simulations, we simulated conditions of the seminal paper by Polsky et al. (

Simulations with the 5-state kinetic NMDAR model generally supported the appropriateness of transfer function assumptions, provided a sufficiently large slope for Mg^{2+} blockade. As stated previously, NMDAR bistability relies on additional currents such as inward-rectifying K^{+} (Shoemaker, ^{2+} blockade, a system composed solely of leak and NMDAR currents will possess a single equilibrium (Figure ^{2+} blockade slope). Simulations over a 50 ms period produced maximum peak EPSP's with the empirically observed “linear-hook” form described previously (Figure ^{2+} midpoint near spike threshold. A simple and more accurate solution for the distributed closing time models would be use of a piecewise function making peak EPSP the maximum of linear (fast ionic) and nonlinear (NMDAR-mediated) components, rather than a bounded sum. Unfortunately, this approach is mathematically undesirable as it does not admit continuous derivatives of all orders. However, burst length distributions add little additional information due to the extremely short spike rise time (Figure ^{2+} blockade. In fact, results indicate that these factors may be exploited to an even greater extent, by further increasing the slope of Mg^{2+} blockade to approach the all-or-none spike threshold near Mg^{2+} blockade's midpoint (Figure

^{2+} blockade necessary for bistability. Like the transfer function only leak and NMDAR currents were considered. ^{2+} blockade slope. To achieve bistability (crossing 0 pA three times) it is necessary to have sufficiently large NMDAR conductance, and Mg^{2+} slope. With increased slope, modest levels of macroscopic conductance permit bistability, while for the standard slope, bistability is not attained for any conductance value.

To test the transfer function's accuracy, simulations were performed under the conditions of Polsky et al. (_{U} = 12 mV, _{L} = −12 mV, α_{L},_{U} = 0.5). The simulation design included two cases of model type (Simple Model and Distribution-based) and both symmetric and asymmetric spatial decay. A functional length constant of 77 μm has been reported for spikes/plateaus in basal dendrites propagating toward the soma (Major et al., ^{2+} flow). As the distal ends of dendrites are “capped,” there is substantially less attenuation for potentials spreading distally. In both cases, the length constant for contributing to spike generation was half that of the respective functional length constant. Simulated results for the Simple Model (Figures ^{2+} likely mediate the relationship. The model based on the distribution for burst/cluster lengths was slightly better with asymmetric length constants, but still mediocre in both cases (Figures

For a second analysis of transfer function accuracy, we compared simulated results for paired-pulse and single-pulse stimulation as by Polsky et al. (_{U} = 16.5 mV, _{L} = −16.5 mV. To further contrast PPF, synapses primed by the initial pulse are allowed, the previously removed fast components of the inter-opening distribution (the full distribution of Wyllie et al.,

We have defined an artificial and a biophysical transfer function to model dendritic integration. Both functions are based upon sigmoidal opening dynamics of NMDA channels, however the biophysical function supports complex combinations of input, whereas the artificial function is agnostic to input location and simply considers a single nonlinear-component with each input equally weighted. Both transfer functions apply a bounded linear transform to the sum of linear and non-linear components to simulate saturation of the dendritic branch. Unlike many previous two-layer abstractions which describe sigmoidal components (Figure

Due to the spatial decay of post-synaptic signals within a branch, the distance between sites of stimulation is critical for determining the nonlinear threshold. As in Figure

Despite its simplicity, the biophysical transfer function is capable of replicating the sorts of non-linear interactions seen in pyramidal dendrites, which are typically expressed as systems of non-linear differential equations. Although some properties are lost in the use of a time-invariant function, such as capacitive membrane interactions, our replication of Polsky et al.'s findings (^{2+} concentration, which lead to complex interactions in driving force between NMDA spikes and occur over variable time scales due to factors such as release from intracellular stores and pumping back into the extracellular fluid. However, based upon simulation results, the function appears well suited for its intended use in estimating peak somatic EPSP. Accurate modeling of dendritic integration takes on increasing importance as a growing body of evidence points to dendritic roles in areas of joint interest to biophysical and artificial neural network modelers, such as place fields and feature detection (Ujfalussy et al.,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.